Neighborhoods of Riemann curves in complex spaces (Q1907470)

Let \(X\) be a complex compact Riemann surface of genus \(g \geq 2\) embedded into a complex two-dimensional manifold \(M\). Let \(L : = N_{X/M}\) be the normal bundle of \(X\) in \(M\). The topological type of the germ of \(M\) around \(X\) depends only on \(g\) and \(\deg (L)\). However, the holomorphic class of this germ depends strongly on \(X\) and \(L\). The properties of this germ depend strongly of the sign of \(\deg (L)\). The more delicate case (after a striking example found by Arnold) is the case \(\deg (L) = 0\). Here the author (using the classical method of studying for all \(k > 0\) the \(k\)-th order embedding of \(X\) in \(M)\) is able to describe completely (as germ) the families of all germs near the given one, when \(\deg (L) > 2g - 2\). This is possible by the vanishing of suitable cohomology groups. These vanishing are true because \(\deg(L)\) is large. For an approach to similar problems some readers may appreciate \textit{S. Kosarew} [Algebraic geometry, Proc. Conf., Berlin 1985, Teubner-Texte Math. 92, 217-227 (1986; Zbl 0631.32008) or J. Reine Angew. Math. 388, 18-39 (1988; Zbl 0653.14002) and references quoted there].